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Distributed Autonomous Robotic Systems pp 261–274Cite as

A Stochastic Optimal Enhancement of Feedback Control for Unicycle Formations

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Part of the book series: Springer Tracts in Advanced Robotics ((STAR,volume 104))

Abstract

We consider an optimal feedback control approach for multiple nonholonomic vehicles to achieve a distance-based formation with their neighbors using only local observations. Beginning with a non-optimal feedback formation control, each agent determines an additive correction term to its non-optimal control based on an elliptic Hamilton-Jacobi-Bellman equation so that its actions are optimal and robust to uncertainty. In order to avoid offline spatial discretization of the stationary, high-dimensional cost-to-go function, we exploit the stochasticity of the distributed nature of the problem to develop an equivalent estimation problem in a continuous state space using a path integral representation. Consequently, each agent independently computes its optimal feedback control using a discrete-time Unscented Kalman smoother. Our approach is illustrated by a numerical example in which five agents achieve a pentagon with aligned and equal velocities.

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References

  1. Anderson, B., Fidan, B., Yu, C., Walle, D.: UAV formation control: theory and application. In: Blondel, V., Boyd, S., Kimura, H. (eds.) Recent Advances in Learning and Control. LNCIS, vol. 371, pp. 15–34. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  2. van den Broek, B., Wiegerinck, W., Kappen, B.: Graphical model inference in optimal control of stochastic multi-agent systems. Journal of Artificial Intelligence Research 32(1), 95–122 (2008)

    MATH  Google Scholar 

  3. van den Broek, B., Wiegerinck, W., Kappen, B.: Optimal control in large stochastic multi-agent systems. In: Adaptive Agents and Multi-Agent Systems III. Adaptation and Multi-Agent Learning, pp. 15–26 (2008)

    Google Scholar 

  4. Bullo, F., Cortes, J., Martinez, S.: Distributed control of robotic networks: A mathematical approach to motion coordination algorithms. Princeton University Press, Princeton (2009)

    Google Scholar 

  5. Dimarogonas, D.: On the rendezvous problem for multiple nonholonomic agents. IEEE Transactions on Automatic Control 52(5), 916–922 (2007)

    Article  MathSciNet  Google Scholar 

  6. Elkaim, G., Kelbley, R.: A Lightweight Formation Control Methodology for a Swarm of Non-Holonomic Vehicles. In: IEEE Aerospace Conference. IEEE, Big Sky (2006)

    Google Scholar 

  7. Fleming, W., Soner, H.: Logarithmic Transformations and Risk Sensitivity. In: Controlled Markov Processes and Viscosity Solutions, ch. 6. Springer, Berlin (1993)

    Google Scholar 

  8. Freidlin, M.: Functional Integration and Partial Differential Equations. Princeton University Press, Princeton (1985)

    MATH  Google Scholar 

  9. Gelb, A.: Applied Optimal Estimation. The MIT Press, Cambridge (1974)

    Google Scholar 

  10. Goldstein, H.: Classical Mechanics, 2nd edn. Addison-Wesley (1980)

    Google Scholar 

  11. Jadbabaie, A., Hauser, J.: On the stability of unconstrained receding horizon control with a general terminal cost. In: Proceedings of the 40th IEEE Conference on Decision and Control, vol. 5, pp. 4826–4831. IEEE, Orlando (2001)

    Google Scholar 

  12. van Kampen, N.G.: Stochastic Processes in Physics and Chemistry, 3rd edn. North Holland (2007)

    Google Scholar 

  13. Kappen, H.: Linear Theory for Control of Nonlinear Stochastic Systems. Physical Review Letters 95(20), 1–4 (2005)

    Article  MathSciNet  Google Scholar 

  14. Kappen, H.J.: Path integrals and symmetry breaking for optimal control theory. Journal of Statistical Mechanics, Theory and Experiment 2005, 21 (2005)

    Article  MathSciNet  Google Scholar 

  15. Kappen, H.J., Gómez, V., Opper, M.: Optimal control as a graphical model inference problem. Machine Learning 87(2), 159–182 (2012)

    MATH  MathSciNet  Google Scholar 

  16. Kushner, H.J., Dupuis, P.: Numerical Methods for Stochastic Control Problems in Continuous Time, 2nd edn. Springer (2001)

    Google Scholar 

  17. Long, A.W., Wolfe, K.C., Mashner, M.J., Chirikjian, G.S.: The Banana Distribution is Gaussian: A Localization Study with Exponential Coordinates. In: Proceedings of Robotics: Science and Systems, Syndey (2012)

    Google Scholar 

  18. Milutinović, D.: Utilizing Stochastic Processes for Computing Distributions of Large-Size Robot Population Optimal Centralized Control. In: Proceedings of the 10th International Symposium on Distributed Autonomous Robotic Systems, Lausanne, Switzerland (2010)

    Google Scholar 

  19. Oksendal, B.: Stochastic Differential Equations: An Introduction with Applications, 6th edn. Springer, Berlin (2003)

    Book  Google Scholar 

  20. Palmer, A., Milutinović, D.: A Hamiltonian Approach Using Partial Differential Equations for Open-Loop Stochastic Optimal Control. In: Proceedings of the 2011 American Control Conference, San Francisco, CA (2011)

    Google Scholar 

  21. Parker, L.E.: Multiple Mobile Robot Systems. In: Sciliano, B., Khatib, O. (eds.) Springer Handbook of Robotics, ch. 40, pp. 921–941. Springer (2008)

    Google Scholar 

  22. Ren, W., Beard, R.: Distributed consensus in multi-vehicle cooperative control: Theory and applications. Springer, New York (2007)

    Google Scholar 

  23. Ryan, A., Zennaro, M., Howell, A., Sengupta, R., Hedrick, J.: An overview of emerging results in cooperative UAV control. In: 2004 43rd IEEE Conference on Decision and Control, vol. 1, pp. 602–607 (2004)

    Google Scholar 

  24. Särkkä, S.: Continuous-time and continuous-discrete-time unscented Rauch-Tung-Striebel smoothers. Signal Processing 90(1), 225–235 (2010)

    Article  MATH  Google Scholar 

  25. Tanner, H., Jadbabaie, A., Pappas, G.: Coordination of multiple autonomous vehicles. In: IEEE Mediterranean Conference on Control and Automation. IEEE, Rhodes (2003)

    Google Scholar 

  26. Todorov, E.: General duality between optimal control and estimation. In: 47th IEEE Conference on Decision and Control, vol. 5, pp. 4286–4292. IEEE, Cancun (2008)

    Google Scholar 

  27. Todorov, E.: Efficient computation of optimal actions. Proceedings of the National Academy of Sciences of the United States of America 106(28), 11,478–11,483 (2009)

    Google Scholar 

  28. Wang, M.C., Uhlenbeck, G.: On the theory of Brownian Motion II. Reviews of Modern Physics 17(2-3), 323–342 (1945)

    Article  MATH  MathSciNet  Google Scholar 

  29. Wiegerinck, W., van den Broek, B., Kappen, B.: Optimal on-line scheduling in stochastic multiagent systems in continuous space-time. In: Proceedings of the 6th International Joint Conference on Autonomous Agents and Multiagent Systems, p. 1 (2007)

    Google Scholar 

  30. Wiegerinck, W., Broek, B., Kappen, H.: Stochastic optimal control in continuous space-time multi-agent systems. In: 22nd Conference on Uncertainty in Artificial Intelligence, Cambridge, MA (2006)

    Google Scholar 

  31. Yong, J.: Relations among ODEs, PDEs, FSDEs, BDSEs, and FBSDEs. In: Proceedings of the 36th IEEE Conference on Decision and Control, pp. 2779–2784. IEEE, San Diego (1997)

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA, 95064, USA

    Ross P. Anderson & Dejan Milutinović

Authors
  1. Ross P. Anderson
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  2. Dejan Milutinović
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Corresponding author

Correspondence to Ross P. Anderson .

Editor information

Editors and Affiliations

  1. Mechanical Engineering and Mechanics, Drexel University, Philadelphia, USA

    M. Ani Hsieh

  2. Department of Mechanical Engineering, Johns Hopkins University, Baltimore, Maryland, USA

    Gregory Chirikjian

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Anderson, R.P., Milutinović, D. (2014). A Stochastic Optimal Enhancement of Feedback Control for Unicycle Formations. In: Ani Hsieh, M., Chirikjian, G. (eds) Distributed Autonomous Robotic Systems. Springer Tracts in Advanced Robotics, vol 104. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55146-8_19

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  • DOI: https://doi.org/10.1007/978-3-642-55146-8_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-55145-1

  • Online ISBN: 978-3-642-55146-8

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